76.13.56 problem 65

Internal problem ID [17559]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 4. Second order linear equations. Section 4.3 (Linear homogeneous equations with constant coefficients). Problems at page 239
Problem number : 65
Date solved : Thursday, March 13, 2025 at 10:12:40 AM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} x^{2} y^{\prime \prime }+3 x y^{\prime }+5 y&=0 \end{align*}

With initial conditions

\begin{align*} y \left (1\right )&=1\\ y^{\prime }\left (1\right )&=-1 \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 13

ode:=x^2*diff(diff(y(x),x),x)+3*x*diff(y(x),x)+5*y(x) = 0; 
ic:=y(1) = 1, D(y)(1) = -1; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = \frac {\cos \left (2 \ln \left (x \right )\right )}{x} \]

✓ Mathematica. Time used: 0.02 (sec). Leaf size: 14

ode=x^2*D[y[x],{x,2}]+3*x*D[y[x],x]+5*y[x]==0; 
ic={y[1]==1,Derivative[1][y][1] ==-1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {\cos (2 \log (x))}{x} \]

✓ Sympy. Time used: 0.198 (sec). Leaf size: 10

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + 3*x*Derivative(y(x), x) + 5*y(x),0) 
ics = {y(1): 1, Subs(Derivative(y(x), x), x, 1): -1} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {\cos {\left (2 \log {\left (x \right )} \right )}}{x} \]